There is known a control apparatus for a vehicle that includes (i) a drive power source and (ii) a mechanically-operated transmission mechanism which constitutes a part of a power transmitting path between the drive power source and drive wheels of the vehicle and which is configured to establish each of a plurality of gear positions by engagement of at least one of a plurality of engagement devices including a one-way clutch. A vehicle control apparatus disclosed in JP-2011-199959A is an example of this type of control apparatus. This document discloses a technique relating to a regenerative torque generated in a vehicle having a motor/generator serving as drive power source, particularly, a technique of controlling the regenerative torque such that, when a shift-down action of a mechanically-operated transmission mechanism of the vehicle is executed in response to reduction of running speed of the vehicle during a coast regeneration (e.g., electric power regeneration performed by the motor/generator during decelerating run of the vehicle without an acceleration operation member and a brake operation member being operated), the regenerative torque is restrained from being increased until the shift-down action is completed, if the brake operation member is operated.